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In this paper, we analyze and discuss the well-posedness of two new variants of the so-called sweeping process, introduced by Moreau in the early 70s (Moreau in Sém Anal Convexe Montpellier, 1971 ) with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving convex subset C ( t ) , supposed to have a bounded variation, by a Lipschitz mapping. Under some assumptions on the data, we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation. The second variant, for which a large analysis is made, concerns a first order sweeping process with velocity in the moving set C ( t ) . This class of problems subsumes as a particular case, the evolution variational inequalities [widely used in applied mathematics and unilateral mechanics (Duvaut and Lions in Inequalities in mechanics and physics. Springer, Berlin, 1976 ]. Assuming that the moving subset C ( t ) has a continuous variation for every t ∈ [ 0 , T ] with C ( 0 ) bounded, we show that the problem has at least a Lipschitz continuous solution. The well-posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator. We also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model (Krejčı in Eur J Appl Math 2:281–292, 1991 ), to the planning procedure in mathematical economy (Henry in J Math Anal Appl 41:179–186, 1973 and Cornet in J. Math. Anal. Appl. 96:130–147, 1983 ), and to nonregular electrical circuits containing nonsmooth electronic devices like diodes (Acary et al. Nonsmooth modeling and simulation for switched circuits. Lecture notes in electrical engineering. Springer, New York 2011 ). The theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like other papers in this collection, we show in this presentation how elegant modern convex analysis was influenced by Moreau’s seminal work.

This paper is devoted to the study of a new class of implicit state-dependent sweeping processes with history-dependent operators. Based on the methods of convex analysis, we prove the equivalence of the history/state dependent implicit sweeping process and a nonlinear differential equation, which, through a fixed point argument for history-dependent operators, enables us to prove the existence, uniqueness, and continuous dependence of the solution in a very general framework. Moreover, we present some new convergence results with respect to perturbations in the data, including perturbations of the associated moving sets. Finally, the theoretical results are applied to prove the well-posedness of a history-dependent quasi-static contact problem.

The new lower tungsten divertor of EAST uses a right-angle shape consisted by horizontal and vertical targets, which has the capacity of increasing the divertor closure. The strike point (SP) sweeping experiment is carried out to (1) avoid long-term deposition of particle and heat flux at the same location, thus protecting the target, (2) study the dependence of power control capability on the SP location. The particle and energy flux densities to the target depends strongly on their decay width. Therefore, it is important to know how the SP location influences the outer target (OT) particle parallel λjs ,OT and OT parallel heat flux decay widths λq ,OT. In this work, SOLPS-ITER simulations combined with SP sweeping experiment are applied to study this issue. Four cases, which are taken from different time during SP sweeping (including both horizontal and vertical divertor) in L-mode experiment with high heating power, are selected for investigation. The simulation result is in satisfactory agreement with experiment data, suggesting the simulation is valid. The results indicate that the SP location can affect neutral particles accumulation and ionization positions, thus affecting λjs ,OT and λq ,OT. (1) When SP is located in horizontal target, the higher neutral particle ionization in common flux region leads to wider λjs ,OT than those of vertical target. (2) When SP is located on horizontal target, the divertor power radiation is higher than that of vertical target, resulting in wider λq ,OT. (3) Increasing upstream plasma density can effectively broaden λq ,OT, while λjs ,OT remains almost unchanged. This study improves the understanding of the influence of divertor shape on λjs ,OT and λq ,OT, and can be applied to heat flux control during long-pulse high-power discharges on EAST.

We present existence of solution and necessary conditions for an optimal control problem with a particular case of sweeping processes with constant sweeping set. Using a sequence of approximating problems we prove existence of solution and necessary conditions. The special feature of our approach resides on the nature of the approximating problems: in contrast with other literature these problems do not rely on distance functions to the sweeping set.

We generalize a maximum principle for optimal control problems involving sweeping systems previously derived in de Pinho et al. (Optimization 71(11):3363–3381, 2022, https://doi.org/10.1080/02331934.2022.2101111) to cover the case where the moving set may be nonsmooth. Noteworthy, we consider problems with constrained end point. A remarkable feature of our work is that we rely upon an ingenious smooth approximating family of standard differential equations in the vein of that used in de Pinho et al. (Set Valued Var Anal 27:523–548, 2019, https://doi.org/10.1007/s11228-018-0501-8).

We consider the Volterra integro-differential equation with a time-dependent prox-regular constraint that changes in an absolutely continuous way in time (a Volterra absolutely continuous time-dependent sweeping process). The aim of our paper is twofold. The first one is to show the solvability of the initial value problem by setting up an appropriate catching-up algorithm (full discretization). This part is a continuation of our paper (Bouach et al. in arXiv: 2102.11987. 2021) where we used a semi-discretization method. Obviously, strong solutions and convergence of full discretization scheme are desirable properties, especially for numerical simulations. Applications to non-regular electrical circuits are provided. The second aim is to establish the existence of optimal solution to an optimal control problem involving the Volterra integro-differential sweeping process.

We show that sweeping processes with possibly non-convex prox-regular constraints generate a strongly continuous input-output mapping in the space of absolutely continuous functions. Under additional smoothness assumptions on the constraint we prove the local Lipschitz continuity of the input-output mapping. Using the Banach contraction principle, we subsequently prove that also the solution mapping associated with the state-dependent problem is locally Lipschitz continuous.

We propose a full study and methodology for multi-view stereo reconstruction with performance capture data. Multi-view 3D reconstruction has largely been studied with general, high resolution and high texture content inputs, where classic low-level feature extraction and matching are generally successful. However in performance capture scenarios, texture content is limited by wider angle shots resulting in smaller subject projection areas, and intrinsically low image content of casual clothing. We present a dedicated pipeline, based on a per-camera depth map sweeping strategy, analyzing in particular how recent deep network advances allow to replace classic multi-view photoconsistency functions with one that is learned. We show that learning based on a volumetric receptive field around a 3D depth candidate improves over using per-view 2D windows, giving the photoconsistency inference more visibility over local 3D correlations in viewpoint color aggregation. Despite being trained on a standard dataset of scanned static objects, the proposed method is shown to generalize and significantly outperform existing approaches on performance capture data, while achieving competitive results on recent benchmarks.

This survey article addresses the class of continuous-time systems where a system modeled by ordinary differential equations is coupled with a static or time-varying set-valued operator in the feedback. Interconnections of this form model certain classes of nonsmooth systems, including sweeping processes, differential inclusions with maximal monotone righthand side, complementarity systems, differential and evolution variational inequalities, projected dynamical systems, and some piecewise linear switching systems. Such mathematical models have seen applications in electrical circuits, mechanical systems, hysteresis effects, and many more. When we impose a passivity assumption on the open-loop system, and regard the set-valued operator in the feedback as maximally monotone, we obtain a set-valued Lur'e dynamical system. In this article we review the mathematical formalisms, their relationships, main application fields, well-posedness (existence, uniqueness, continuous dependence of solutions), and stability of equilibria. An exhaustive bibliography is provided.